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ister represented by noninteracting many-body (multi- theinteractionmatrixelementsU becomecomparableto |
qubit) states of ideal qubits. A simple estimate [24] the level spacing between many-body states ∆ . How- |
n |
shows that the residual interaction J will be unavoid- ever, in nature we have only two-body interaction and |
ably much larger than the energy level spacing ∆ be- therefore, while the size of the Hilbert space grows ex- |
n |
tween adjacent eigenstates of a realistic/generic quan- ponentially with the number of particles n, the number |
tum computer . Let us assume that J is relatively weak ofnonzerointeractioninducedmatrixelementsgrowsnot |
comparing to one-qubit level spacing ∆0. Then all N fasterthann2. Tostudythespectralstatisticsinthissit- |
H |
eigenenergies will be located in the energy band of size uation a two-body random interaction model (TBRIM) |
∆E n∆0 and the average multi-qubit level spacing is wasintroducedanditwasshown[29,30]thatinthelimit |
∼ ∆02−n |
∆ ∆E/N ∆0. For the experimental of strong interaction the RMT remains valid even if the |
n H |
≈ ∼ ≪ |
proposals[11,12]wehave∆0 1Ksothatforn=1000, full Hamiltonian matrix is exponentially sparsed. How- |
∼ |
when Shor’s algorithm becomes useful, the multi-qubit ever,muchmoretimewasneededtounderstandthecase |
spacingisincrediblysmall∆ 103 2−103 ∆0 10−298 of relatively weak interaction and to find the critical U |
n c |
∼ × ∼ |
K. This value will definitely be much smaller than any above which quantum chaos and RMT set in. Contrary |
physical residual interaction J. For the proposal [12] to naive expectation U ∆ assumed by many au- |
n |
∼ |
with a distance between nuclear spins of r = 200 ˚A thorsuntilrecently(seee.g.[31–33])itnowbecameclear |
and an effective Bohr radius of a = 30 ˚A ( Eq.(2) that the quantum chaos border is exponentially larger |
B |
of [12]), the coupling between qubits (spin-spin interac- than ∆ since only transitions between directly coupled |
n |
tion) is J ∆0 1 K. By changing the electrostatic states, and not the bulk density of states, are important |
∼ ∼ |
gate potential, an effective barrier between nuclei can for level mixing. As a result the interaction should be |
be modified that can be viewed as a change of effec- larger than the energy level spacing ∆ between directly |
c |
tive electron mass possibly up to a factor of two. Since coupled states U > U ∆ ∆ to change the level |
c c n |
J (r/a )5/2exp( 2r/a )/a ,anda isinverselypro- spacing statistics P(s) f∼ rom th≫ e Poisson distribution to |
B B B B |
∝ − |
portionaltotheeffectivemass,thisgivesaminimalresid- the Wigner-Dyson one and to generate quantum ergod- |
10−5 |
ual spin-spin interaction of J K ∆ . On the icity andchaosforeigenstates. As formy knowledge,for |
n |
∼ ≫ |
first glance this would lead to a natural/naive conclu- the first time this condition for quantum chaos in many- |
sion that at such residual interaction the ideal quantum bodysystemshadbeenformulatedby˚Abergwhoalsoby |
computer eigenstates are strongly mixed and completely numericalsimulationsfora nucleus modelhadfoundthe |
2 |
changein the levelstatistics acrossthis border[34]. Due particle states is ∆2 V/m2, ∆3 V/m3 respectively |
to that I will call this condition ˚Aberg criterion. with ∆ ∆2 ∆3∼ . The two-b∼ ody matrix elements |
≫ ≫ |
In spite of other ˚Aberg’s papers (e.g. [35]) his result written in the noninteracting eigenbasis are supposed to |
was not broadly known for the community probably be- berandomwithatypicalvalueU12 U andU23 U for |
∼ ∼ |
causehisstudiesweremainlyaddressedtonuclearphysi- interaction between 1st/2d and 2d/3d particles respec- |
cists and the numerical results were not so confirma- tively. At the same time the interaction matrix element |
tory at that time. Also in nuclei the interaction is usu- U13 between1st/3distakentobezeroforsimplicity(the |
ally quite strong and the quantum chaos sets in rather final result remains the same for U13 U). For two in- |
∼ |
quickly. In fact it was shown by numerical simulations teracting particles, e.g. 1st/2d, the levels become mixed |
of other models that in an isolated many-body Fermi by interaction and RMT spectral statistics sets in when |
system a sufficiently strong interaction can induce dy- the interactionbecomes largerthanthe levelspacingbe- |
namical thermalization with the Fermi-Dirac distribu- tween two-particles states: U12 > ∆2. On the contrary |
tion [31,32]. Independently , the line of research related for U12 ∆2 the perturbation theory is valid and the |
≪ |
to the problem of two interacting particles in a random eigenstateswithinteractionaredeterminedbyonenonin- |
potential [36] showed that a two-body interaction can teractingeigenstate. Inthecaseofthreeparticlesthesit- |
lead to a number of unexpected results and that disor- uationismorecomplicatedsincethethree-particlestates |
der/chaoscanstronglyenhancetheinteraction. Thelast are not directly coupled by two-body interaction. |
property in fact had been known from the spectacular The matrix element U3 between 3-particle levels can |
Sushkov-Flambaum enhancement of weak interaction in be found by the second order perturbation theory. In |
nuclei [37]. The analytical studies of few particles with this way the matrix element between initial state 123> |
′ ′ ′ | |
two-bodyrandominteractionU [38]showedthatthemix- andfinalstate 123 > is givenby diagrampresentedin |
inter| 1′¯23>. |
ing of levels, quantum chaos and RMT statistics appear Fig.1 with mediate state |
| |
only if U > ∆2, where ∆2 ≫ ∆3 ≫ ∆4... are the level <12U12 1′¯2><¯23U23 2′ 3′ > U2 |
spacing between 2-,3-,4-particle levels respectively. This U3 =X | | | | (1) |
(E1+E2+E3 −E1′ −E¯2 −E3) ∼ ∆ |
result was generalized for many particles and the quan- ¯2 |
tum chaos border U ∆ was proposed and confirmed |
c ∼ c It is important that the summation is carried out only |
byextensivenumericalstudiesoftheTBRIMmodel[40], oversingle particle states ¯2 andthe sumis mainly deter- |
independently of ˚Aberg’s papers. As it will be seen from |
minedbyatermwithaminimaldetuningindenominator |
the present paper the knowledge obtained for quantum |
being of order ∆. As a result the level mixing sets in for |
many-body systems can be successfully used for such a |
U3 >∆3 that gives the quantum chaos border [38]: |
new direction of research as quantum computing. |
InthispaperIreviewtherecentdevelopmentsofquan- U U c ∆2 ∆3 (2) |
∼ ∼ ≫ |
tum chaos theory in many-body systems obtained in |
Thismeansthatthe3-particlelevelsaremixedonlywhen |
Toulouseandshowtheirlinksandimportanceforaquan- |
theinteractionmixestwo-particlelevelsthatistheconse- |
tumcomputerwhichalsorepresentsamany-bodysystem |
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